3.30 \(\int \frac{1}{-5-3 \cos (c+d x)} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{2 d}-\frac{x}{4} \]

[Out]

-x/4 + ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])]/(2*d)

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Rubi [A]  time = 0.0127493, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2658} \[ \frac{\tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+3}\right )}{2 d}-\frac{x}{4} \]

Antiderivative was successfully verified.

[In]

Int[(-5 - 3*Cos[c + d*x])^(-1),x]

[Out]

-x/4 + ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])]/(2*d)

Rule 2658

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, -Simp[x/q, x] - Sim
p[(2*ArcTan[(b*Cos[c + d*x])/(a - q + b*Sin[c + d*x])])/(d*q), x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && NegQ[a]

Rubi steps

\begin{align*} \int \frac{1}{-5-3 \cos (c+d x)} \, dx &=-\frac{x}{4}+\frac{\tan ^{-1}\left (\frac{\sin (c+d x)}{3+\cos (c+d x)}\right )}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0259359, size = 20, normalized size = 0.65 \[ \frac{\tan ^{-1}\left (2 \cot \left (\frac{1}{2} (c+d x)\right )\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(-5 - 3*Cos[c + d*x])^(-1),x]

[Out]

ArcTan[2*Cot[(c + d*x)/2]]/(2*d)

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Maple [A]  time = 0.029, size = 18, normalized size = 0.6 \begin{align*} -{\frac{1}{2\,d}\arctan \left ({\frac{1}{2}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-5-3*cos(d*x+c)),x)

[Out]

-1/2/d*arctan(1/2*tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 2.12741, size = 32, normalized size = 1.03 \begin{align*} -\frac{\arctan \left (\frac{\sin \left (d x + c\right )}{2 \,{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*cos(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*arctan(1/2*sin(d*x + c)/(cos(d*x + c) + 1))/d

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Fricas [A]  time = 1.61944, size = 72, normalized size = 2.32 \begin{align*} \frac{\arctan \left (\frac{5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/4*arctan(1/4*(5*cos(d*x + c) + 3)/sin(d*x + c))/d

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Sympy [A]  time = 0.779307, size = 44, normalized size = 1.42 \begin{align*} \begin{cases} - \frac{\operatorname{atan}{\left (\frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{2} \right )} + \pi \left \lfloor{\frac{\frac{c}{2} + \frac{d x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor }{2 d} & \text{for}\: d \neq 0 \\\frac{x}{- 3 \cos{\left (c \right )} - 5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*cos(d*x+c)),x)

[Out]

Piecewise((-(atan(tan(c/2 + d*x/2)/2) + pi*floor((c/2 + d*x/2 - pi/2)/pi))/(2*d), Ne(d, 0)), (x/(-3*cos(c) - 5
), True))

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Giac [A]  time = 1.13732, size = 41, normalized size = 1.32 \begin{align*} -\frac{d x + c - 2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-5-3*cos(d*x+c)),x, algorithm="giac")

[Out]

-1/4*(d*x + c - 2*arctan(sin(d*x + c)/(cos(d*x + c) + 3)))/d